The study of rogue wave patterns is important as this information allows for the prediction of later rogue wave events from earlier wave forms. Although graphs of low-order rogue waves have been plotted for many integrable equations, and simple patterns such as triangles and rings have been reported, richer patterns arising from high-order rogue wave solutions have received little attention. For the NLS equation, preliminary investigations on rogue wave patterns were made in [1]}, [2]}, [3]} through Darboux transformation and numerical simulations. It was observed in [1]} that if a \(N\) -th order rogue wave exhibits a single-shell ring structure, then the center of the ring is a \((N-2)\) -th order rogue wave. This observation was explained analytically in [2]}. In [3]}, it was observed that NLS rogue patterns could be classified according to the order of the rogue waves and the parameter shifts applied to the Akhmediev breathers in the rogue-wave limit. This latter observation allowed the authors to extrapolate the shapes of rogue waves beyond order six, where numerical plotting of rogue waves became difficult. However, an analytical and quantitative prediction of NLS rogue patterns at arbitrary orders is still nonexistent.
